Why does the sun appear brightest during final moments of a total eclipse?

Don't do it! It's actually easier to ruin your vision by watching the solar eclipse than watching the sun. The surface brightness is every bit as bright but the total brightness is lower, the sky is darker, and it tricks your eye into opening up and recieving more light.

The macula has most of your good vision, and that's the part that gets damaged.

Always use a neutral density 5 filter until the filter turns black after the diamond ring, and then use the filter again a few seconds before 3rd Contact.

The iris of the eye is adjusting so it looks that much brighter. Too, atmospheric pollution plays something of a role.